Current-driven electromagnetic ion cyclotron instability at substorm onset

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Current-driven electromagnetic ion cyclotron instability

at substorm onset

S. Perraut, O. Le Contel, A. Roux, A. Pedersen

To cite this version:

S. Perraut, O. Le Contel, A. Roux, A. Pedersen. Current-driven electromagnetic ion cyclotron

in-stability at substorm onset. Journal of Geophysical Research Space Physics, American Geophysical

Union/Wiley, 2000, 105 (A9), pp.21097-21107. �10.1029/2000JA900059�. �hal-03086730�

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 105, NO. A9, PAGES 21,097-21,107, SEPTEMBER 1, 2000

Current-driven electromagnetic ion cyclotron instability

at substorm

onset

S. Perraut, O. Le Contel, and A. Roux

Centre d'ttude des Environnements Terrestre et Planttaires, CNRS/UVSQ, V•lizy, France A. Pedersen

Department of Physics, University of Oslo, Oslo, Norway

Abstract. ULF waves at frequencies of the order of the proton gyrofrequency are systematically detected at the early development of substorm breakups. The observed characteristics of these

ULF waves,

namely

their polarization

and fiE/fiB

ratio are consistent

with being

electromagnetic

waves driven unstable by a parallel current. In order to take into account properly wave particle interactions, a kinetic approach is used. We show that a parallel drift between electrons and ions leads to a strong instability, resulting from a coupling between the shear Alfvtn (SA) mode and the fast magnetosonic mode via this drift. We call it current-driven Alfvtn instability (CDA). We have carried out a parametric study of this current-driven electromagnetic instability in a

parameter range adapted to conditions prevailing at the geostationary orbit before and during breakup. We conclude that even a modest parallel drift between electrons and ions (Vd), caused by a parallel current, can alestabilize CDA waves. When the ratio between Vd/V A (V A being the Alfvtn velocity) increases, the CDA mode couples with SA mode. These two modes have a substantial parallel electric field that leads to a fast parallel diffusion of the electrons. We suggest that this parallel diffusion leads to an interruption of the parallel current.

1. Introduction

The signature on particles instruments and fluxgate magnetometer of a substorm with dispersionless injected

particles, occurring while the European geostationary spacecraft GEOS 2 was close to the magnetic equator, has been described by

Roux et al. [1991a, b] and ?erraut et al. [1995]. These studies provide evidence for the formation of thin current sheets, even as close from the Earth as the geostationary orbit [Lui et al., 1977].

Similar observations have been carried out on AMPTE/CEE by

Lui et al. [1992]. For the present study we have selected one substorm event: January 25, 1979 (a detailed description of field and particle measurements for this event is given by Roux et al. [1991a, b] and Perraut et al. [1993]) that occurred while GEOS was 3 ø off the geomagnetic equator. In addition to particle and fluxgate magnetometer measurements, we use wave

measurements from the electric field instrument and from the

low-frequency triaxial search coil magnetometer. Robert et al.

[1984] have already pointed out that large-amplitude

electromagnetic fluctuations are observed at substorms with most

of the energy being below FH+, the proton gyrofrequency. The frequency range, howewer, largely exceeds FH+. In this frequency

range one expects to find fast magnetosonic (FMS hereafter) waves and therefore expects the polarization to be right- handed/compressional (depending on the direction of the wave normal) and the 1•SEI/I/SBI ratio to be of the order of the Alfvtn

Copyright 2000 by the American Geophysical Union. Paper number 2000JA900059.

0148-0227/00/2000JA900059509.00

velocity

VA

(more

precisely

Vn•-•[)-•Fa+)

:r, where

F is the

emission frequency). Pertaut et al. [1993] found that the polarization characteristics and the 1tSEI/ltSBI ratios are not consistent with the waves being FMS waves. One possibility, put forward by Perraut et al., is that the mode is the intermediate mode studied by Formisano and Kennel [1969] with a bifluid approach. Yet the bifluid approach is not necessarily valid especially at and above F,•+, then a kinetic approach is needed to resolve this issue. Given that the observed waves are strongly related to breakup and could therefore play a role at triggering substorms, we have carded out a more detailed study with more refined tools, in particular we use the kinetic theory to identify the modes and reanalyze GEOS 2 data.

In section 2, we present the main observations underlying the present study; we characterize waves around F•,+. Section 3 describes the solutions of the dispersion relations of low- frequency waves in a large 15 plasma (15 is the ratio between the kinetic and the magnetic energies, ~1) obtained with a kinetic approach. We have chosen a set of parameters that describe the plasma conditions encountered in the plasma sheet. We show that the presence of field-aligned electrons leads to the development of a strong instability resulting from a coupling (via electrons drifting along field lines with respect to ions) between the FMS mode and the shear Alfvtn (SA) mode. In section 4, the kinetic theory is used for a parametric study of this electromagnetic current-driven instability. The results obtained for different drift velocities, and magnetic field amplitudes (or 15) are compared with observations. Section 5 is devoted to the coupling between the electromagnetic current-driven instability and the SA mode. Finally, the propagation of the waves along the field lines is briefly discussed, and a scenario is proposed to explain the role that these waves could play at substorms.

21,098 PERRAUT ET AL.' CURRENT-DRIVEN ION INSTABILITY AT SUBSTORM ONSET 2. Characterization of the ULF Waves Observed at

Substorms and Free Energy Source

The selected event corres[x)nds to dispersionless injection of energetic elecmms and ions. The lack of energy dispersion (not shown) suggests that the acceleration is a local process. The three components V, D, and H of the dc magnetic field B. displayed in Figure l a, give key intbrmation about the magnetic configuration. A large V component (in this case V=H prior to breakup) indicates that a thin current sheet has developed and that GEOS, though being close to magnetic equator, is not at the center of this current sheet. The increase of the H component is a clear indicator of the dipolarization of the B field. The integrated [x)wer of the magnetic fluctuations up to 10 Hz are also displayed in Figure 1. A detailed analysis of the spectra of the electric and magnetic components of the Pil has been reported by Perraut et ai. 11993l tk•r this event. These waves have a broad band spectrum observed in the same frequency range as Pil oscillations observed on the ground IHeacock, 1967, Biisinger et ai., 19811. Figure 1 shows (1) that while the dipolarization develops, -1 rain period oscillations (identified as ballooning modes of the thin current sheet [Holter et al., 19951) with periods in the Pi2 range superiml•)se on the quasi-do increase of H and (2) that higher~

frequency transient fluctuations (F~ F,+) develop simultaneously

in the Pil range. The occurrence of waves in the Pil range is a salient feature of substorm breakup. This is regularly observed, at least for dispersionless injections events ]Perraut et ai., 1998 I.

The /k)rmation of thin current sheets during substorm growth phase has been studied by Le Contel et al. [1998, 2{XX)I. It has been shown that time variations associated with the fi•rmation of a current sheet impose, via the quasi-neutrality, a field-aligned

electric field. Such an electric field will accelerate electrons along the field lines, either equatorward or toward the ionosphere, depending on the ltxzal time variation of the perturbation iml•)sed by the solar wind and on the direction of the pressure gradient. This parallel electric field can produce the free energy source that drives waves unstable. The parallel potential drop found by Le Contel et al., is of the order of I(X} V. There is a parallel current asstx:iated with this parallel electric field; the corresponding drift velocity between electrons and ions should be of the order of a t•w 1½) • m/s, which is difficult to measure. Whenever a sufficient pitch angle coverage was available, field-aligned electrons were found together with waves IShepherd et al., 1990; Perraut et al., 19981. at breakup. This suggests that field-aligned low-energy electrons are related to the intensification of the ULF waves. Similar observations have been made by Johnswne et ai. 119941, who have also observed field-aligned bursts of electrons in the low energy range, often associated with changes in the configuration of the magnetic field. These intensifications were found by Johnswne et ai,. to be generally bidirectional. but fluxes depend on the latitude of the spacecraft and are not the same in the parallel and antiparallel directions: hence there is a current. These observations and the corresponding increase in the field- aligned electron flux are consistent with the presence of a field- aligned drill of elecm)ns with respect to ions that can drive waves unstable. Motivated by these results, we have solved numerically the full electromagnetic, linear dispersion relation for Mawellian electrons streaming with a drift velocity V,, (along the magnetic field B) with respect to Maxwellian ions, in a homogeneous, infinite Vlasov plasma. Like Gary. et ai. [19761 and Forslund et ai. [ 19791. we are considering a finite 13 plasma.

120 I::: 80

ß

-•

40

c:

0

10 •

A

0_.

•

0-3

10 -5

0.5<f<

1Hz

'

ß

(UT) 19:45

20 '15

20:45

Figure 1. (a) V, D, and H components (V is radial from the Earth, D is azimuthal, and H orthogonal to V and D) and total dc magnetic field B. (b) Integrated power of B components of waves around F,+, between 0.5 and 11 Hz. All

these parameters have been measured on GEOS 2 on January 25, 1979. Notice the simultaneity between the

occurrence of the "high-frequency waves" around the proton gyrofrequency, the fluctuations of the dc magnetic

PERRAUT ET AL.' CURRENT-DRIVEN ION INSTABILITY AT SUBSTORM ONSET 21,099

3. Electromagnetic Current Driven Instability

In the equatorial region where our observations took place. the parameter 13 is typically ~1. Gary and Forslund [19751. Gat'), et ai. [19761 and Forslund et al. [19791 have shown that. in a large 13 plasma, the usual current-driven instability (see, for instance, Drummond and Rosenbluth 11962]) is modified. As soon as

13>m/m,. a new electromagnetic instability is found. with a velocity threshold comparatively (i.e., in terms of the ratio V,.•/V•) lower than the corresponding electrostatic instability. Thus. even a modest drift of electrons relative to the ions (along a dc magnetic field) may excite waves around the ion gyrofrequency. with a finite magnetic coml•)nent.

In view of the observations described above. the electromagnetic current-driven instability is a gtx)d candidate to account for wave observations. We have solved the kinetic dispersion relation with a software (WHAMP) designed by R•hmmark [19821. for a set of parameters that fits the observations. The density is 1 cm. the magnetic field is 50 nT (giving F,+-0.76 Hz and V,,~I.I(X) km/s), and the plasma is composed of two populations, ions and electrons with MaxwellJan distributions and a thermal energy of 12 keV for the ions, and 2 keV for the electrons (T,/T•=6, typical for the plasma sheet). In these conditions. 13- 1. Since the parallel propagation (0=0 ø) has been already studied by Gary et al. [19761. oblique propagation (0=45 ø) is mainly considered in this work (0 denotes the angle between the wave vector and the magnetic field). The effect of a differential drift velocity between the ions and the electrons is studied by comparing the solutions with those obtained in the simple case where there is no drift.

Figure 2 displays the dispersion relation t•)r V,,•=(}, 0.03 and (). 1 V,, (where V,, is the thermal velocity of the electrons: thus V, corresponds to O, 2, and 20 eV. respectively) and for 0=45 ø. The frequency •2•:, normalized to the proton gyrofrequency and the normalized growth rate ht/f•.+ are plotted as functions of the normalized wave number K=kV,/f•.+. Similar solutions have been obtained by Gary et al. 119761 for 0=0 ø with a different ratit) T,/T• (=0.5 in their case).

For Va =0 one gets four solutions: two branches with positive frequencies corresponding respectively to FMS (labeled plus) and SA (labeled minus), while for negative frequencies: (plus) corresponds to SA and (minus) to FMS. The notation (plus or minus) is the same as in the work of Gary et al. 119761. the reason for its use will be clarified later. The SMS mode is damped to be followed. Comparison between Figure 2a and the Figure 1 from Gary et al. [19761 shows that the variation of the frequency of the SA mode as a function of K, strongly depends on the propagation angle. With our set of parameters and for parallel propagation (not shown) the normalized frequency reaches a maximum around 0.25 for K~0.9, while for 0=45 ø (Figure 2a) the frequency monotonously increases with K, as expected for the intermediate mode in a high 13 plasma and for oblique propagation [Formisano and Kennel, 19691. All modes are damped when Va =0 (Figure 2a').

As soon as a small parallel drift of electrons relative to the ions is introduced an instability with a large growth rate develops (see Figures 2b, 2b', 2c and 2c') for 0=45 ø and Figure 2 in Gat3, et al. [ 19761 for 0=0ø). Again, the behavior depends on the propagation angle:

1. For parallel propagation, in a warm plasma. the number of solutions does not change {Gary et al., 1976, Figure l i: the frequencies of solutions labeled plus are slightly modified with respect to the case V,, =0, while solutions labeled minus have a very different sha•; they are no longer symme•ic with res•ct to the other r{x•ts. and merge t•r small K (kpt<().3). This merged branch and the adjacent branch with a •)sitive frequency, t•,r 0.3< kp,<l, exhibit a large positive growth rate (Figure 2 of et ai. 119761).

2. For oblique propagation one gets solutions that ltx)k like the

case Kt =0 and 0=45 ø (see Figure 2 ), but there is a striking

difference: the ap•ance of a new root at low frequencies with a large •sitive growth rate. Hereafter we will call the branch with a •sitive growth rate T: CDA, standing for current-driven Alfv6n wave. To identify the nature of the corres•)nding instability. we will comp•e the kinetic solutions to an analytical approximation obtained in the bifluid approach.

The current-driven electromagnetic instability has also •en studied by Hasegawa 119751. A general dispersion relation is given by his formula (2.5()). As the dis•rsion relation is not solvable analytically in the general case. we have considered a simple situation where ions are "cold" and station•y whereas the electrons am hot with their thermal vel{•ity much l•ger than the phase velt•ity and drift at a velocity V• with respect to the ions. Thus the ions can be represented by the delta thnction and we can assume a drifting Maxwellian distribution for the electrons. If we choose a coordinate system fixed to the ions the dis•ibution functions in the p•allel direction are given by

f'" = •(V...) o II

where V::, and Vz, •' •e the p•allel thermal vek•ities of ions and

electrons. Thus the two dispersion relations D+ and D- for p•allel propagation write:

'

{) (l)

I 2 ((0-

kV,,

)

f•

2•,,

co

D-:

k2c

2

_(02

_ (0•,,•

kV,

e Z,(•')

+

0 (2)

where the arguments •.2 of the plasma dispersion function Z•.2

[Fried and Come. 1961 ! •e

f•p,,

t0•.,

f•,,+ and

•.•, are the ion and

electron

plasma

frequencies

In the plasma conditions under consideration co<<m.,, thus. as

long as k<<co,.,11, 5, •1,2 >>l. and the asymptotic expansion of Z(•)

leads to

D+:

k2c

2 (0

2

+W•,e

((0-k,V•) 2 0

D- k-'c:

(02

+(0),•

((0-k,V,) , to

(0

- t•

. , V

, +(0.

A tbrmula similar to (4) is given by Hasegawa [1975]. At low

frequency.

for parallel

propagation

and for fIp./fI.+>>l, these

21,100 PERRAUT ET AL.' CURRENT-DRIVEN ION INSTABILITY AT SUBSTORM ONSET 0 0.5 1 1.5 2 kV/f• +

0=45 [

0 ... :;; ... : ...

i

...

G-,

= -0.25

...

:

...

: ...

:

/

_+.=

iI

..i.

- _ ..i.. '!

•

0 ....

-'•:---•----:

... •.--•--- /: ...

q. i

i

^ .

+ 0.5

Figure

2. Warm

plasma

dispersion

relations

for

(a,

a') V•0, (b,

b') V•O.O3V,

e,

and

(c,

c') V•0.1V,e,

for

oblique

propagation

(0=45ø).

The

normalized

fre.quencies

•fl.+ and

normalized

growth

rate

),/fl.+

are

plotted

versus

normalized

K, defined

as K=kVA/I2.+.

B=50

nT, T?12

keV,

T•=2

keV,

N,=N•=I

cm

'3,

B=8•NKBT/B

2 -1. Two

symmetric

couples

of waves

are

found

for V•0: For

positive

frequencies,

plus

is the

FMS

mode,

minus

is the

SA

mode.

In Figures

2b, 2b',

2c,

and

2c',

the

solid

curves

correspond

to the

kinetic

solutions

and

the

dashed

curves

are

the

solutions

of the

approximate

equations

(5') and

(6').

A new

branch

called

CDA

mode

is strongly

amplified

for

V•0.03

V,,

and

for V•=0.1V,,.

Notice

the

substantial

positive

growth

rate

for the

CDA mode.

The

other

modes

have

a

negative

),/I2.+.

The

approximation

provides

a relatively

good

determination

of the

CDA mode

for the

real

as well

PERRAUT ET AL.: CURRENT-DRIVEN ION INSTABILITY AT SUBSTORM ONSET 21,101

D+' X

z-

X(K•2•

+

K,•-•)-(K•2•

+

K,•-•)=0 (5)

D-

: X

2

+

X

(

K

,•-,

- K

, •-•

) - (

K

,

z,

-K,

V-•-)=0,

(6)

where K and X are the normalized variables defined above,

X=• f2.+ and K= k. V,/f2.+.

In the case of nonparallel propagation approximate formulas ((5') and (6') below) can be found. Equation (5') is deduced from

(5) by replacing

K//•

by K: in D+. One

can

verify

that

for V

d

=0 the

positive roots of D+ and D- correspond to the FMS and SA modes, respectively. Comparison with exact solutions shows the usefulness of the approximate formulas (5') and (6').

D+' X

•-

-X(K

• +

K,•df)-(K

• +K,v

•) =0 (5')

O-' X2+

X(K,

2,

- Kn

v•)-(K,2

, - K//

•) =0. (6')

Equations (5) and (6) have two solutions each. For 0=0 ø. the roots (plus) are weakly modified by the drift. and the solutions of (5) fit pretty well with the exact solutions given by Ga.rv et al. [ 1976]. The roots (minus), howewer, merge (complex conjugate) to give the CDA mode. Analytical solutions of (6) also fit the numerical ones [Gary et at, 1976].

For 0=45 ø, (5') and (6') also have two solutions each. Again, the roots (plus) (dashed lines) fit well the exact result (solid curves), as can be seen in Figure 2b, 2b', 2c, and 2c'. The roots (minus) also fit well with the numerical solutions at least for

K<0.75 (for V, =0.03V,•) and for K<2.5 (for V, =0.1V,•). The

number of roots found numerically, howewer, depends on Va. For small V d (V d =0.03) the "classical" SA mode found for Vd =0 is still present, in addition to the CDA mode (hence there are five roots for a given K). For larger V d the classical SA disappears (too damped).

Although the topology of the branches plus has changed, they can be identified by following the progressive deformation of the solutions with increasing Va and by comparison with the analytical expression. The approximation reproduces relatively well the shape of the real part of solutions + as well as the CDA mode. However while the CDA mode (which only exists in the presence of a drift) is strongly unstable, the classical Alfv6n mode and FMS modes are damped.

Let us now focus on the only unstable mode: the CDA mode. Given that the analytical expression describes relatively well this mode, we will use it to ease the discussion.

The discriminant intervening in the solution of (6') is null for four values of K: K,=0, K,=V,•/V, and

1Va

I 16Vn2

=

l-

).

When

V) > 16V,•

2, K,,,.

2 is real

and

the

wave

is unstable

in two

K,,

domains: O<K,,<K,,• and K,a <K,,< V•/V A which explains why the

growth rate can present two maxima for a set of parameters. Notice that positive growth rates are only obtained for K,, < VdlV A.

In the opposite

case where

V•

2 <16V,,

•, one can neglect

K/,(K/?Va/VA) with respect to 4. Thus an approximate solution of

(6') can be found; it writes

= g.(•-fn-g.)_+i -g..

(7)

The slope of the real part for small K, is OX/OK,? V, /2 V,, which gives in the (X, K,) frame (V• cos 0)/2V,, as can be verified from

Figure 2. Although (7) is approximate, it gives the correct

threshold in K,, namely the minimum V•IV A, for getting a positive

growth rate

The CDA mode identified in these conditions is an interesting new feature; while the classical FMS or SA modes are absorbed

('l'/f2.+

< 0), the electron

drift produces,

for small

K,,, a strong

instability; 'l'/f2.+ reaches -0.3, even for a low value of

(0.03

V,) corresponding

to a bulk

electron

energy

of-2 eV (Figure

2). We will now develop a parametric study of the role of the drift

velocity

Va,

of the magnetic

field

B 0 and

of the ion temperature

T,

on the frequency and the growth rate of the CDA instability.

4. Parametric Study of the CDA Mode

The propagation angle 0, the relative drift velocity between ions and electrons V,, and the Alfv6n velocity are the main parameters

in (6'). We have found that the frequency and the growth rate

depend very little upon the electron temperature; therefore the instability has a fluid type, a conclusion consistent with the approximate dispersion equation (6) obtained for parallel propagation. Thus the growth rate depends mainly on the drift

velocity of the electrons and not on the slope of the electron

distribution at least as long as V,<<V,e. Notice, however, that ions

are damping the wave that explains why a kinetic approach is needed. We study the behavior of the kinetic solution as a function of 0 and then solve the dispersion relation for selected values of 0 and for different V• and 13.

4.1. Effect of the propagation angle 0

Even without drift velocity between ions and electrons the

shape of the dispersion relation changes very much with 0. For

example, for 0>40 ø the frequency of the SA mode increases monotonously, instead of increasing until it reaches a maximum

(generally below F,+) and then decreasing as K increases.

Therefore it is interesting to study the modification of the

dispersion relation of the CDA mode as a function of 0 for a given drift velocity V, =0.2V,•, which corresponds to an energy of

80 eV. Figure 3 displays the variations of a full set of parameters

characterizing the waves ((õElõB, bE,/õE ratios, group velocity...) for different angles of propagation. Figure 3a shows that for K<2

the reduced frequency depends linearly on K//, more precisely the

normalized phase velocity at low frequency is (V, cos 0)/2V, as

discussed above. The maximum frequency corresponds to

K-O.5V,/V A (.•2.4 for the set of parameters selected here). The growth rate has a more complex behavior (Figure 3b): for small propagation angles a relative minimum occurs between the two

maxima at K~2, as expected from the approximate solutions of (6'). This minimum however disappears for 0>40 ø. As a consequence, for oblique propagation (0-45ø), one expects to get a wide band spectrum coveting a frequency range >3F.+. For K<I and K>3 the growth rate decreases with 0. For 0-0 ø the mode corresponding to a maximum growth rate at small K is identified as "kink-like instability" by Gary et al. [1976], while

the mode corresponding to the maximum at large K is identified

as "whistler current instability."

The parallel group velocity is generally larger than the

perpendicular group velocity, at least for K<2.5 which

corresponds to the most unstable mode (compare Figures 3d and

3e). In the same range of K, the õE/fiB ratio is larger than the

Alfv6n velocity and does not depend strongly upon 0 (Figure 3c).

The parallel electric field is finite: bE,/-O.O2õE•. (Figure 3f), and sharply increases up to 0.8•5E for K-5, but this range of K

corresponds to values where the CDA mode is damped. Notice that for K-5 the group velocity has a nonnegligible perpendicular

21,102 PERRAUT ET AL.' CURRENT-DRIVEN ION INSTABILITY AT SUBSTORM ONSET .

0

2

4

6

0

.... 2 ...

4

6

A .

•

kV

/f•+ •

1

0.5 ...

! ...

+ 0.5

0 0 .

0

2

4

6

0

2

4

6

kV/f• +

kV /f• +

A H

•

A H

•

0.6

...

i ...

!'-75 ...

0.2;

...

! ...

i

...

0

2

4

6

0

2

4

6

A H

•

kV/f•

+

Figure

3. Normalized

frequencies,

growth

rates,

fiE/fiB

ratios

(compared

to

VA),

;SEJISE,

VgJVg

....

and

Vga./Vg,o

, for

the

CDA

mode

as

a function

of normalized

K for

a fixed

V

d (V•=

0.2

V,•),

13-1

and

for

different

propagation

angles.

The

other

parameters

are

the

same

as

in Figure

2; •f•.+ and

?/f•.+

are

maximum

for

0=0

ø,

and

the

wave

is well

guided

along

the

magnetic

field

for K<2 and

in the

opposite

direction

above

(K>2).

The

dispersion

relation

exhibits

a

maximum

frequency

at K~O.5VJVA=

2.4.

The

phase

speed

at low

frequency

is (Vd

cos

0)/2

VA.

component and that the fiE/fiB ratio vanishes. For 0=0 ø, there is

no parallel

electric

field

component

and

the

perpendicular

group

velocity

is null (as expected).

Since

the presence

of a parallel

electric

field is a critical

parameter

for studying

wave-particle

interactions,

we will carry

out

the

rest

of the

parametric

study

of

the dispersion relation (with WHAMP) for an intermediate

angle,

0=45%

corresponding

to a finite

E,.

4.2. Effect of the drift velocity

The angle of propagation is now fixed to 45 ø , and four different

values of V• are considered in Figure 4. The larger the drift velocity, the higher is the emitted frequency (Figure 4a). When

the

drift

velocity

increases,

the growth

rate

becomes

large

over

a

wider

range

of K and

exceeds

0.8 for V• =0.2

V,•

(Figure

4b). For

PERRAUT ET AL.: CURRENT-DRIVEN ION INSTABILITY AT SUBSTORM ONSET 21,103

larger V a, for example, V a =0.3 V,,, the growth rate has two maxima

and is positive over a wide range of K. Notice that for an even

larger V d (V d > 0.5V,•, not shown here), the range of K corresponding to wave growth (positive ¾) reduces. A large drift

velocity (V a > 0.1) destabilizes frequencies below and above the proton gyrofrequency. We have found that the CDA mode is LH

polarized and strongly compressional whatever V a and 0. The

larger V a, the larger is bE/fiB (Figure 4c); for instance, bE/bB=3

for K=3 and V a =0.3V,,. The wave is well guided along the dc magnetic field for different ranges of K depending on V a (Figure

4d), for a small

K: k, Vg

>0, while

for high

K (K> 5): k, Vg<0.

For

K<2 (Figure 4d). The bEJbEratio (Figure 4f) reaches -0.02 for

1 0 2 4 6 kV12 + 1.5 1

>• 0-

-0.5'

.

45

E

o.5'.

•" -1

....

i

...

-2 ... ! ... ...

Va=

0.03}

V,e i 0=45

...

:

0.1

'

0.2

^.

:

02

0=45 (•.1

;0.2

0.•

0.2

...

i

....

kV12

+ •

kV12

+ •-•

Figure 4. Same as Figure 3, for 0 fixed (0=45ø), and 13-1, for different drift velocities Vd between ions and electrons. ogf•.+, 'y/f•.+ and bE/bB increase with Va. The range of K where ¾ is positive also increases with Va. Notice that bEJbE =0.02 over the unstable range of K. See text for discussion.

21,104 PERRAUT ET AL.: CURRENT-DRIVEN ION INSTABILITY AT SUBSTORM ONSET

l<K<4 and sharply increases above 0.7 for K~2.5 (for V d = 0.1V,e) and 5 (for Vd = 0.2 V,,). These large increases occur near the value of the K, corresponding to the marginal stability.

4.3. Role of the plasma parameter g

During substorm growth phase the intensity of the magnetic field can change by a factor of 2-3 near the equator, in the near-

Earth plasma sheet (NEPS) (see, for example, Figure 1, where B o

is -50 nT, whereas its typical value is 100 nT at the geostationary orbit). Furthermore, B 0 increases along the field line, which has to be taken into account for studying wave propagation along the field lines. Then it is very important to consider the modification of the dispersion relation as a function of B 0 (or 13), and to keep the density constant. Figure 5 displays the corresponding results. The reduced frequency (Figure 5a), the growth rate (Figure 5b), and the bE/bB ratio (Figure 5c) increase when 13 increases (or B 0

12

1.5

...

i !• ' 4

+

o.54

...

i ...

-X

...

!

...

•

>• 0 ...

-0.5

kV

/f2

+

•

kV

/f2

+

•

:i 0=45

•" 0

0'5

i

0-4

:>-0.5•

i 1 '

0=45

2'---

':

...

}

0.6

...

! 1 ...

0.4-

l

0.7.

.... i ...

Figure 5. Same as Figure 3 for 0 fixed (0=45 ø) and VA= 0.2V,,, for different plasma parameters 13. When 13 increases

(B o decreases), oYf2.+, ¾/f2.+ and bE/fSB are increasing. The range of K where the wave is well guided is reduced

PERRAUT ET AL.: CURRENT-DRIVEN ION INSTABILITY AT SUBSTORM ONSET 21,105 0.4 0.3 ÷

0.2-

,

Vd= .07

Vex

s^rx,

0.2

•

:

:

0.064

-/---•

...

r '- • '' r

.... i .... i .... l .... i

ß Vd=

0.055

Vte

0.3.

S-t/ 0=45

•=: 0.2-

Vd•

0.064

V

t

s^ ',

.... I .... i .... I ....

A .

Vd=,0.064

Vt,

'

....

A .

Figure 6. (a, b, c) Dispersion relations of CDA and SA modes for 0=45 ø and for slightly different V d (Vd cos 0 -VA). The other parameters are the same parameters as in Figure 2. (d) Growth rate and (e)parallel electric component for

V•= 0.064V,

e, corresponding

to the electron

drift velocity

when

the two modes

couple.

Notice

the strong

parallel

decreases for a fixed temperature). Thus the decrease of the magnetic field associated with the building up of a thin current sheet is especially favorable for getting waves with high

frequencies (above F.+), large growth rates and large fiE/fiB

ratios. Moreover a high 13 gives a wider range of K with large parallel group velocities (Figure 5d). As seen previously,

the 15E,fiSEratio reaches -0.02 for a wide range of K and

maximizes just below the frequency for marginal stability (Figure 5f).

5. Coupling Between CDA and SA Modes

The CDA mode is well guided along the magnetic field at least for low values of K. During their propagation toward the

ionosphere the CDA waves will encounter a region where

Va

cos

0 -V A.

This

leads

to a coupling

between

the CDA and

SA

modes for a given propagation angle, as illustrated in Figure 6. Figures 6a, 6b, and 6c display the dispersion relation of the CDA and SA modes for slightly different values of Va/V,e. For 0=45 ø

21,106 PERRAUT ET AL.' CURRENT-DRIVEN ION INSTABILITY AT SUBSTORM ONSET

and V d =0.064V, e (Figure 6b) the two dispersion curves merge. Thus one expects that a fraction of the wave energy generated in the CDA mode will mode convert into the SA mode and keep on

propagating earthward in that mode. This is consistent with the

observations made at high latitude by the Interball spacecraft, which show very similar spectra [Perraut et al., 1998]. Figures 6d

and 6e display the growth rate and parallel electric component of these two modes. The CDA mode is still unstable, while the SA mode is damped via Landau resonance with electrons. Notice that the SA mode has a much larger (10 times) parallel electric field. These remarks can help understanding the role of these waves on substorm development. The CDA waves that are easily destabilized in the vicinity of the equatorial plane can propagate along the field line and reach a region where they are convened into SA waves. These SA waves have a strong parallel electric field (kinetic Alfvtn waves, see Goertz [ 1986] and Hasegawa and Uberoi [1982]) and larger phase velocities, hence they can heat

the electrons, via Landau resonance, as discussed in the next

section.

6. Propagation Along Field Lines

A parallel current that is symmetric with respect to the magnetic equator, cancels at the equator and its modulus increases with the latitude. Thus V d/Vt•, and therefore the growth rate, tend to increase with the latitude. On the other hand, B decreases very fast with the latitude, which leads to a reduction in the growth rate. Thus the growth of CDA waves will take place at low latitudes, but off equator. Once generated CDA waves are well guided along the field lines. For an earthward parallel drift (Vd >0) the phase velocity of CDA waves is also earthward (co/k,>0), but the group velocity can be either positive or negative, depending on k. As illustrated in Figure 7, CDA waves generated

with small

k will propagate

earthward

(co/k,,

and V•, >0), while

CDA waves

generated

with large

k will propagate

with V•,<0,

equatorward, untill they are reflected. Figure 7, curve A, shows the dispersion relation, off equator, in the maximum growth rate region. As the wave propagates towards higher latitudes

0.4. , , .

0.3

SA i ?

! CDAW

0

014

0.8

Figure 7. Dispersion relations for CDA and SA modes for two

different sets of parameters corresponding to two different positions along a field line, the abscissa is the parallel normalized k vector, k•. has been fixed along the field line (k•.=0.63). At point A: B=40 nT, V• =872 km/s; point B: B=55 nT, V• =1200 km/s. At

point B the CDA wave converts into SA wave. The SA mode is

later strongly damped, while its phase velocity increases (not

shown). See the text for discussion.

(dispersion relation labeled B), F is constant, F.+ increases, and K,/=k, Va/f2.+ decreases (V•/f•+ is assumed to be constant). The corresponding motion in the (F, k,) plane is indicated by an arrow. Curve B is defined by V• cos 0 -Va; at this point the SA and CDA modes merge (see the discussion in the previous

section).

Then a fraction of the energy will be convened into SA mode. Since the SA is also a guided mode it will keep on propagating toward higher latitudes; the corresponding dispersion relation is indicated by dotted line. During this propagation/mode conversion, F is fixed and k, decreases, hence •k, increases. Therefore we expect that the electrons will be accelerated/heated via Landau resonant interaction with the CDA/SA modes.

For the calculations described above, drifting electrons have been assumed to move only in one direction. If a symmetric beam is assumed, instead, and thus if there is no net current, wave

growth disappears, at least above Fi_i+. Even a slight asymmetry

either in the density or in the velocity (a field-aligned current), howewer, leads to recover a large growth rate. Such an asymmetry has been reported by Johnstone et al. [1994]. The magnetic signature seen on the V and D components in Figure 1 during early breakup (2017-2018 UT), also suggests the existence of a field-aligned current.

7. Conclusion

In an attempt to interpret ULF waves observed at breakup, we have carried out a parametric study of electromagnetic current- driven Alfvtn (CDA) waves. This mode results from the coupling between the FMS and SA modes via the relative parallel drift between electrons and ions (the parallel current). Thus the CDA mode has mixed properties; it is left-handed polarized as the SA mode and compressional, as the FMS. The instability is very robust; it is found to develop over a wide range of parameters. Present results are consistent with those of Gary et al. [ 1976], but we focus on nonparallel propagation and on values of T•/T•>>I, consistent with observations in the plasma sheet. The dispersion relation of CDA waves is different from that of the classical FMS/SA modes, in particular, for small K, it writes o•k,V d/2, therefore the parallel phase velocity is proportional to V• and not to Va. While SA and FMS modes are damped, even a small drift velocity destabilizes CDA waves. For oblique propagation the growth rate is about as large as for parallel propagation, and the frequency range is wider. The frequency range and ratios fiE/fiB critically depend on B; for the large B met in the equatorial NEPS, and for Vd/Vt•_>O. 1 and oblique propagation the frequency range

extends above F.+ and the •E/fB ratio is a few times V,•.

Therefore the observations of waves above F.+ does not imply that they are propagating in the FMS mode. The wide frequency

range (beyond F.+), and the large values of fiE/fiB obtained here

fit with the observations described by Perraut et al. [ 1993]. Given the strong relation between CDA waves and breakup, one is tempted to think that these waves play a role in accelerating particles and/or triggering substorms. Let us describe a possible scenario. As discussed in the previous subsection, the growth rate of CDA waves takes place at low latitudes, but off equator. Once generated CDA waves are well guided along the field lines. As they travel to higher latitude, CDA waves undergo a mode coupling with the SA waves, as illustrated in Figure 6. Thus the fraction of the energy, which is convened into SA mode, will propagate earthward (see Figure 7), essentially at the Alfvtn velocity. Since the Alfvtn velocity increases rapidly away from the magnetic equator, it will match the electron thermal velocity

PERRAUT ET AL.: CURRENT-DRIVEN ION INSTABILITY AT SUBSTORM ONSET 21,107

(co/k,,--V•--V,). Then CDA/SA waves that have a finite parallel electric field will be strongly damped via Landau damping. For to<to,.,, the first adiabatic invariant is conserved for electrons, thus electron diffusion will be along the magnetic field. We will study. in a forthcoming paper. the parallel diffusion of electrons and its effect on the reduction of the parallel current. Then a microscopic process: a current-driven instability is likely to have consequences on large-scale configuration of the tail via a modification of the current system.

Acknowledgements. We acknowletlge useful discussions with G.

Belmont. This work has been supported by CNES and by GDR PLASMAE

and PNST of CNRS.

Michel Blanc thanks S. Peter Gary anti another referee lbr their assistance in evaluating the paper.

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(Received October 29, 1999; revised March 17, 2000;